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This is just a question about notation, but it got no useful answers on math.stackexchange.

Let $L$ be the Lie algebra of $n\times n$ Hermitian matrices, with Lie bracket $(A,B)\mapsto i(AB-BA)$.

In a context where $A$ and $B$ are understood to be elements of $L$, I'd like to write $[A,B]$ for the commutator of $A$ and $B$, but I'd also like to write $[A,B]$ for the Lie bracket of $A$ and $B$. Obviously, because the commutator and the Lie bracket are not equal, I can't do both.

Is there a well-established standard about what the symbol $[A,B]$ means in this context?

(Note: Dirac, in his papers on quantum mechanics, uses $[A,B]$ to mean the Lie bracket, not the commutator. But I don't want to assume that the notation used by physicists in the 1930s is standard among mathematicians in the 21st century.)

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    $\begingroup$ This is one of the common confusions caused by the physicists' habit of treating the space Hermitian matrices as a Lie algebra rather than the mathematically more natural space of anti-Hermitian matrices as a Lie algebra (using the commutator). Of course, since simply multiplying by $i$ is a linear isomorphism from Hermitian matrices to anti-Hermitian matrices, one can induce the physicist' Lie algebra structure on the Hermitian matrices by declaring this to be a Lie algebra homomorphism. It's really just a matter of convention (though, sometimes, signs cause problems in the translation). $\endgroup$ Aug 11, 2015 at 21:54
  • $\begingroup$ @RobertBryant: Thanks for this --- but I'm not sure it answers the question. In a math paper, is it less confusing to use $[A,B]$ for the commutator, or for the Lie bracket, or ought one not use this notation at all? $\endgroup$ Aug 11, 2015 at 22:02
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    $\begingroup$ In a math paper, when the space is the anti-Hermitian matrices, we use the commutator $[,]$ and know that it is a Lie bracket. Mostly (probably, there are some exceptions), mathematicians don't regard the Hermitian matrices as a Lie algebra, so there is no confusion. In the rare cases when a mathematician wants to do something like this, he or she makes an explicit (re-)definition of the symbol and warns the reader of the nonstandard usage. (Frequently, the referee objects anyway.) $\endgroup$ Aug 11, 2015 at 22:10
  • $\begingroup$ @RobertBryant: This sounds like it is probably the answer I'm looking for --- though, because there are two possible non-standard usages, I'd still like to know whether one is generally preferred to the other. $\endgroup$ Aug 11, 2015 at 22:29
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    $\begingroup$ I can't help you there. I never use either of these nonstandard conventions; they are equally bad from my point of view. $\endgroup$ Aug 11, 2015 at 22:32

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The notation $[x,y]$ for a Lie bracket is fairly standard, although there are books on Lie algebras using $xy$ instead. Indeed, for matrices the notation $[A,B]$ may also denote the commutator $AB-BA$. This can be confusing if the subspace of matrices does form a Lie algebra, but not with respect to the standard commutator, e.g., for $\mathfrak{su}(2)$ with respect to the Hermitian Pauli matrices; or for other cases, where the Lie bracket is given more generally by $$[A,B]:=AQB-BPA$$ for specific matrices $P$ and $Q$, not necessarily the identity matrix, see here.
Suppose we have a Lie algebra $L$ of matrices where $AB-BA$ does not define a Lie brakcet. Then we might pass to an isomorphic Lie algebra $L'$ in some $\mathfrak{g}l_n(K)$ by Ado's theorem, so that $AB-BA$ in $L'$ defines a Lie bracket (see also here).

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  • $\begingroup$ Actually, the paper you link to (in the answer you link to) denotes the non-standard Lie bracket $(A,B)$, not $[A,B]$. $\endgroup$ Aug 14, 2015 at 23:31

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